Download - Section 2.7
Use Absolute Value Functions and Transformations
Section 2.7
In Lesson 1-7, you learned that the absolute value of a real number x is defined as follows.
You can also define an absolute value function
, if is positive0, if 0
, if is negative
x xx x
x x
.f x x
Parent Function for Absolute Value Functions
The parent function for the family of all absolute value function is f (x) = |x|.The graph of f (x) = |x| is V-shaped and is symmetric about the y-axis. So, for every point (x, y) on the graph, the point (-x, y) is also on the graph.The highest or lowest point on the graph of an absolute value function is called the vertex.The vertex of the graph f (x) = |x| is (0, 0).
To the left of x = 0, the graph is given by the line y = -x.
To the right of x = 0, the graph is given by the line y = x.
(-2, 2) (2, 2)
vertex
(0, 0)
Transformations
A transformation changes a graph’s size, shape, position, or orientation.A translation is a transformation that shifts a graph horizontally and/or vertically, but does not change the size, shape, or orientation.The graph y = |x – h| + k is a graph of y = |x| translated h units horizontally and k units vertically.The vertex of y = |x – h| + k is (h, k).
y = |x|
(0, 0) h
k(h, k)
y = |x − h| + k
Graph y = | x – 1 | + 3. Compare the graph with the graph of y = | x |.Comparisony = | x – 1 | + 3 translate y = | x | one unit to the right and three units up.
Example 1
-5 -4 -3 -2 -1 1 2 3 4 5
-4
-3
-2
-1
1
2
3
4
Stretches and Shrinks
When |a| ≠ 1, the graph of y = a|x| is a vertical stretch or a vertical shrink, depending on whether |a| is less than or greater than 1.
For |a| > 1 For |a| < 1• The graph is
vertically stretched or elongated
• The graph is vertically shrunk or compressed
• The graph of y = a|x| is narrower than the graph of y = |x|.
• The graph of y = a|x| is wider that the graph of y = |x|.
When a = -1, the graph of y = a|x| is a reflection in the x-axis of the graph of y = |x|.When a < 0 but a ≠ -1, the graph of y = a|x| is a vertical stretch or shrink with a reflection in the x-axis of the graph of y = |x|.
Reflections
Compare each graph with the graph y = | x |.Comparison(a) y = 1/3| x | is the graph of y = | x | vertically
shrunk by a factor of 1/3
Example 2
1Graph a and b 2 .3
y x y x
-5 -4 -3 -2 -1 1 2 3 4 5
-4
-3
-2
-1
1
2
3
4
(b) Comparison:y = -2| x | is the graph of y = | x | vertically stretched by a factor of 2 and reflected over the x-axis.
-5 -4 -3 -2 -1 1 2 3 4 5
-4
-3
-2
-1
1
2
3
4
A graph may be related to a parent function using multiple transformations.The graph of y = a|x – h| + k can involve a vertical stretch or shrink, a reflection, and a translation of the graph of y = |x|.
Multiple Transformations
Compare the graph with the graph of y = | x |.ComparisonThe graph of y = ¼| x + 3 | − 2 is the graph of y = | x | first vertically shrunk by a factor of ¼ then translated 3 units to the left and 2 units down.
Example 31Graph 3 2.4
y x
-8 -6 -4 -2 2 4 6 8
-6
-4
-2
2
4
6
A landscaper sketches the design for a triangular shrub protector on graph paper. Write an equation for the shrub protector.
Example 4
2 4 6 8 10 12 14 16-1
1
2
3
4
5
6
7
8
9
O
The vertex is (5, 6).A point on the graph is either (0, 0) or (10, 0). Now solve for a.
5 6y a x
0 0 5 6a
1.2a
1.2 5 6y x
Transformations of General Graphs
The graph of y = a ∙ f (x – h) + k can be obtained from the graph of any function y = f (x) by performing these steps:1. Stretch or shrink the graph of y = f (x)
vertically by a factor or |a| if |a| ≠ 1.If |a| > 1, stretch the graph.If |a| < 1, shrink the graph.
2. Reflect the resulting graph from #1in the x-axis if a < 0.
3. Translate the resulting graph from #2 horizontally h units and vertically k units.
The graph of a function y = f (x) is shown. Sketch the graph of the given function.
Example 5
(-3, -3)
(0, 0)
(3, -6)
1a3
y f x
(-3, -3)
(0, 0)
(3, -6)
(-3, 1)
(3, -2)
b 1 3y f x
(-3, -3)
(0, 0)
(3, -6)
(-2, 0)
(1, 3)
(4, -3)